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Abstract

"This paper is a report of an investigation of an approximate method for finding the principal mode frequency roots for beams in transverse vibration. The method utilizes approximate transmission matrices obtained by a power series expansion of the basic differential equation which governs the transmission matrix. Investigation has been carried out to examine the efficiency of the method in producing the first several normal mode frequency roots. This has been achieved by applying the method to several uniform and non-uniform beams and comparing the results with the exact solutions. The technique has further been applied to non-uniform beams of tapered rectangular cross section and the results are given in non-dimensional form for use in practical application. Frequency root errors obtained by this method when applied to uniform beams of Fixed-Fixed and Fixed-Free ends are seen to be proportional to l/N² (N, the number of segments), when two terms in the series are considered and N is large . With three and four series terms errors are proportional to 1/N³ . The method utilizes only two variables of the beam to be analyzed, the cross sectional area and the area moment of inertia. Progressively better results are obtained by increasing the number of terms in the series--Abstract, Page ii.